A Brief Primer on Ideal Point Estimates

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Dina Robertson
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1 Joshua D. Clinton Princeton University January 28, 2008

2 Really took off with growth of game theory and EITM. Claim: ideal points = player preferences. Widely used in American, Comparative and International Relations: Measure legislator/judicial/voter/party/country preferences, gridlock intervals, uncovered sets, polarization, heterogeneity, representation, party pressure. Outline: Statistical Models al Considerations Reasons for Concern

3 Claim 1: The maps [of ideal points] are useless unless the user understands both the spatial theory that the computer program embodies and the politics of the legislature that produced the roll calls (Poole, 2005).

4 Claim 2: Legislative voting is one of the most intensively studied areas of quantitative political science. Many have assumed that the basic answers were discovered long ago. Others will interpret the recent research as filling in everything they always wanted to know about legislative voting and (often) didn t care to ask. But the challenge of the field remains (Weisberg, 1977).

5 Statistical Model ( ) U it ( θy(t) ) = U it θn(t) = Legislator i votes Yea if f (xi, θ y(t) ) + ζ it f (xi, θ n(t) ) + ν it f (x i, θ y(t) ) f (x i, θ n(t) ) > ν it ζ it Pr(y it = 1) = Pr(ν it ζ it < f (x i, θ y(t) ) f (x i, θ n(t) )) Ideal point estimators vary in assumptions about deterministic parts (f(x i, θ y(t) ),f(x i, θ n(t) )) and stochastic parts (ζ it, ν it ) of legislator i s utility function.

6 NOMINATE & Quadratic Quadratic: f (x i, θ y(t) ) = f (x i, θ n(t) ) = `x i θ y(t) 2 `x i θ n(t) 2 f (x i, θ y(t) ) f (x i, θ n(t) ) = `x i θ y(t) 2 + `x i θ n(t) 2 = β tx i α t Where: α t = θ 2 n(t) θ 2 y(t) and β t = 2 `θ y(t) θ n(t).

7 NOMINATE & Quadratic Quadratic: f (x i, θ y(t) ) = f (x i, θ n(t) ) = `x i θ y(t) 2 `x i θ n(t) 2 f (x i, θ y(t) ) f (x i, θ n(t) ) = `x i θ y(t) 2 + `x i θ n(t) 2 = β tx i α t NOMINATE: f (x i, θ y(t) ) = f (x i, θ n(t) ) = f (x i, θ y(t) ) f (x i, θ n(t) ) = βexp 1 `xi θ 2 y(t) 2 βexp 1 `xi θ 2 n(t) 2 h β exp exp `xi θ y(t) 2 `xi θ n(t) 2 i

8 Stochastic Elements Assume ζ it and ν it are: uncorrelated and drawn from a symmetric, unimodal distribution. Suppose ν it ζ it N(0, σ 2 ): Therefore: Implying: f (x i, θ y(t) ) f (x i, θ n(t) ) N(ν it ζ it, σ 2 ) Pr(y it = 1) = Pr ( ν it ζ it < f (x i, θ y(t) ) f (x i, θ n(t) )) ) = Φ ( σ 1 (f (x i, θ y(t) ) f (x i, θ n(t) ))) ) Pr N (y it = 1) = [ [ Φ β exp ( 12 ω ( ) ) 2 x i θ y(t) exp ( 12 ω ( ) )]] 2 x i θ n(t) Pr Q (y it = 1) = Φ [ σ 1 (β t x i α t ) ]

9 Likelihood Functions Quadratic Model: L(x, θ y) = Π L i=1π T t=1pr Q (y it = 1) y it (1 Pr Q (y it = 1)) 1 y it NOMINATE Model: = Π L i=1π T t=1φ (β tx i α t) y it (1 Φ (β tx i α t)) 1 y it L(x, θ y) = Π L i=1π T t=1pr N (y it = 1) y it (1 Pr N (y it = 1)) 1 y it h = Π L i=1π T t=1φ 1 Φ h β exp h exp h β 1 2 `xi θ y(t) 2 exp 1 `xi θ 2 y(t) 2 exp 1 2 `xi θ n(t) 2 i yit 1 `xi n(t) 2 i 1 yit θ 2

10 Parameter Identification: Clearly unidentified without further restrictions: xβ α = x β α with x = xs and β = β/s Implies L(β, α, x y) = L(β, α, x y) Scale and Rotational Invariance:

11 Parameter Identification: Clearly unidentified without further restrictions: xβ α = x β α with x = xs and β = β/s Implies L(β, α, x y) = L(β, α, x y) Solution: two linearly independent restrictions on x. Sen. Helms = 1, Sen. Kennedy = -1 E[x] = 0, V [x] = 1, and median Democrat < 0.

12 Identification Summary Estimator Comparability Constraints Interest Group Scores Not None WNOMINATE Not None DNOMINATE Across time, not inst. (except pres.) Parametric Change DWNOMINATE Across time, not inst. (except pres.) Parametric Change Common Space Across time and instit. Fixed x Quadratic Depends Depends

13 Snowe Chafee Nelson, D. Breaux Bayh Baucus Lincoln Jeffords Carper Landrieu Pryor Conrad LIEBERMAN Feinstein Nelson, B. Inouye Dorgan Reid Biden Bingaman Kohl Cantwell Schumer Hollings Rockefeller Dodd GRAHAM, B. Byrd Akaka Murray Wyden Leahy Daschle Johnson Stabenow Kennedy EDWARDS Corzine Durbin Feingold Mikulski Lautenberg Dayton Clinton Levin Boxer Harkin KERRY Reed Sarbanes Cornyn Lugar Hatch Sessions Thomas McConnell Allen Cochran Santorum Bunning Grassley Burns Bennett Frist Nickles Craig Alexander Crapo Domenici Bond Enzi Inhofe Kyl Lott Miller Chambliss Voinovich Dole Roberts Allard Talent Brownback DeWine Ensign Hagel Warner Coleman Hutchison Graham, L. Fitzgerald Gregg Shelby Smith Sununu Stevens Murkowski Campbell Collins Specter McCain

14 Snowe Chafee Nelson, D. Breaux Bayh Baucus Lincoln Jeffords Carper Landrieu Pryor Conrad EBERMAN Feinstein Nelson, B. Inouye Dorgan Reid Biden Bingaman Kohl Cantwell Schumer Hollings Rockefeller RAHAM, B. Dodd Byrd Akaka Murray Wyden Daschle Leahy Johnson Stabenow Kennedy EDWARDS Corzine Durbin Feingold Mikulski Lautenberg Dayton Levin Clinton Boxer Harkin KERRY Reed Sarbanes Burns Frist Nickles Craig Alexander Crapo Domenici Bond Enzi Inhofe Kyl Lott Miller Chambliss Dole Voinovich Roberts Allard Talent Brownback DeWine Ensign Hagel Warner Coleman Hutchison Graham, L. Fitzgerald Gregg Shelby Smith Sununu Stevens Murkowski Campbell Collins Specter McCain Cornyn Sessions Thomas Hatch Lugar Cochran McConnell Allen Santorum Bunning Grassley Bennett

15 Much easier now: WNOM9707.EXE (WNOMINATE executable) WNOMINATE for R pscl (ideal) in R TurboADA for STATA (see Jeff Lewis, UCLA) winbugs

16 Recommendations All produce the same answer (in 1d) for well-behaved legislatures. Slightly prefer quadratic for computational reasons: Easier to customize the estimator. Measures of parameter uncertainty are recovered directly. Code is well-understood (and replicable in other programs).

17 Model Evaluation Error structure. Functional form. Goodness-of-fit (degrees of freedom).

18 Exogeneity How to interpret ideal point estimates: Agenda is endogenous & institutional rules change over time.

19 Number of Laws/Votes All Statutes Congress Number of Laws/Votes Top 3500 Statutes Congress

20 Exogeneity How to interpret ideal point estimates: Agenda is endogenous & institutional rules change over time. Decision to vote is endogenous & roll calls not always recorded.

22 Exogeneity How to interpret ideal point estimates: Agenda is endogenous & institutional rules change over time. Decision to vote is endogenous & roll calls not always recorded. Often related to characteristics being explained.

23 Political science is heavily dependant on measures of latent concepts (e.g., ideal points, party manifestos, expert surveys). al issues are largely resolved. Theoretical and interpretative issues remain. Data generating process important (and not well understood).

112th U.S. Senate ACEC Scorecard

112th U.S. Senate ACEC Scorecard HR 658 FAA Funding Alaska Alabama Arkansas Arizona California Colorado Connecticut S1 Lisa Murkowski R 100% Y Y Y Y Y Y Y Y Y S2 Mark Begich D 89% Y Y Y Y N Y Y Y Y S1